NUMERICAL ANALYSIS OF THE MIXED BOUNDARY VALUE PROBLEM FOR THE SOPHIE GERMAIN EQUATION

Abstract

The Sophie Germain equation in variational form is considered on a plane bounded domain under mixed boundary conditions. The original problem is continued to a rectangular domain along the boundary with the Dirichlet boundary conditions. The continued problem is discretized using the finite element method. We present a method of fictitious components, which, for the absolute error, has a convergence rate as a geometric progression. For the numerical solution of the continued problem, we use the method of iterative extensions, which is obtained as a generalization of the method of fictitious components. For the relative error, the method of iterative extensions has the same rate of convergence as a geometric progression in a stronger norm than the energy norm of the problem. The choice of iterative parameters is based on the minimum residual method. An algorithmic implementation of the iterative extension method with automatic selection of iterative parameters is presented. The algorithm also provides a criterion for automatically stopping the iterative process when a given estimate of the relative error is reached. The software implementation of this algorithm does not require additional complications associated with the complexity of the geometry of the domain, the order of the differential equation and the presence of Dirichlet boundary conditions. We consider an example of calculations by the method of iterative extensions within the numerical solution of a particular problem. In order to perform computations, we use the condition of evaluating the relative error in a norm, which is stronger than the norm generated by the problem operator. But the relative errors for the obtained numerical solution to the example of the original problem are considered in other ways. For example, a point-by-point calculation of the relative error is carried out for the grid points. It turns out that several iterations are enough to have a relative error of only a few percent, even with different methods of calculating the error. In practice, computational experiments confirm the optimality of the method indicated in theory.